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June 22, 2011 / marios

The Special Theory of Relativity: A concise explanation

I was not able to find online a simple, concise explanation of Einstein’s Special Theory of Relativity. I am not a physicist, am I not an expert, I might even be naive to be trying this, but here is my take nonetheless:

Consider two coordinate systems:

  1. S: A train station.
  2. T: A train moving with constant velocity v on a straight railway line passing in front of the train station.

There are two postulates:

  1. The laws of nature should have the same expression irrespective of the coordinate system used to derive them.
  2. The speed of light is constant, and equal to c, in vacuum.

The first postulate makes intuitive sense. All coordinate systems should be equivalent for expressing the laws of nature. We should be able to derive exactly the same laws for any natural phenomenon irrespective of if we are observing that phenomenon positioned on the train station or on the train.Why would any particular coordinate system have any particular advantage over expressing the laws of nature in a simpler form?

The second postulate has been shown to be valid through various observations and experiments (here is an argument by De Sitter).

Based on these, let a ray of light travel parallel to the train at the direction of the train, and assume that an observer on the train station determines that the speed of light is c. Then, based on classical mechanics, an observer on the train should determine the speed of light to be c-v, which contradicts postulate 1. Hence, either postulate 1 is not true or classical mechanics is not correct.

Einstein claimed that the latter is true. Here is why:

  1. The relativity of simultaneity: Two strokes of lightning hit points A and B on the railway line simultaneously with reference to an observer standing on the train station. Assume that at the time the strokes hit with reference to the train station, a passenger on the train sits exactly on the midpoint between A and B. Are the strokes of lighting simultaneous for the passenger as well? The answer is no. Since light travels with constant speed c and the passenger travels towards point B with speed v, as far as the passenger is concerned a stoke of lightning hits B before another stroke of lighting hits A. Hence, events that are simultaneous with reference to the train station are not simultaneous with reference to the train. Every coordinate system has its own particular time.
  2. The relativity of distance: Consider two points A and B on the train, travelling with velocity v. We can measure the exact distance of A and B while travelling on the train using a tape. We can also measure the distance between A and B from the train station, using the same tape, by simply determining two points A’ and B’ on the rails that are being just passed by A and B at a particular time t with respect to the train station. Suppose that the distance of A and B on the train is w. Then, is it necessary that the distance of A’ and B’ on the train station is also w? Lets assume that it is NOT necessary.

Let the coordinates of an event with respect to S be x, y, z, t. Then, based on classical mechanics, the coordinates of the same event with respect to T are x’ = x – vt, y’=y, z’=z, t’=t.

But, based one the two observations above, if we allow time and distance to be relative, what are the values x’, y’, z’, t’ of the same event with respect to T such that the law of the transmission of light in vacuum is satisfied with respect to both S and T? The values can be expressed using a system of equations known as the “Lorentz Transformation“:

x' = \frac{x - vt}{\sqrt{1 - \frac{v^2}{c^2}}}, y' = y, z' = z, t' =  \frac{t -x\frac{v}{c^2}}{\sqrt{1-\frac{v^2}{c^2}}}

This transformation between the two coordinate systems does not violate postulate 1. A light ray travelling parallel to the rails advances in accordance with x=ct, and by substituting into the Lorentz transformation for x’ and t’, we get x’=ct’. The transformation was specifically derived so that it will satisfy the constancy of the propagation of light.

What are the consequences of this transformation?

  1. Distance contraction: Let the distance of points A and B be 1 meter on the train. Based on the Lorentz transformation we will find that the distance of A’ and B’ is . The distance has contracted as far as the observers on the train station are concerned. Nevertheless, as far as the passengers are concerned, everything is fine. Distance is relative!
  2. Time dilation: Similarly, consider a clock permanently fixed at position x’=0 of T. Let t’=0 and t’=1 be two consecutive ticks of the clock. As judged from the passengers on the train the time that elapses between the two ticks is 1 second. As judged by the observers on the train station, by using the Lorentz transformation, the time that elapses between the two consecutive ticks is  seconds. That is, a somewhat larger time. Time is relative!

To recap, if we assume that postulates 1 and 2 make sense, then classical mechanics cannot be correct. Nevertheless, there is a simple framework (called the Special Theory of Relativity) based on the Lorentz transformation that can satisfy both postulates. All we have to do is to alter our beliefs that distance and time are absolute across all coordinate systems.

Notice that the special theory of relativity talks only about coordinate systems that move with uniform velocities with respect to each other (no acceleration, no circular motions, etc.). Einstein believed that postulate 1 should be true irrespective of the relative movement of coordinate systems. Hence, he had to come up with the general theory of relativity, but that is a different beast.

Numerous people have come up with examples that seem to lead to contradictions based on the special theory of relativity. Einstein and others were able to show that the special theory of relativity was consistent and did not lead to any contradictions in all cases. The most famous example is the twin paradox, and Wikipedia has a nice simple example. I would recommend reading it, but just to give an intuition, either twin can apply the Lorentz transformation and assume that the other twin is travelling in the opposite direction. Hence as far as either twin is concerned, the other twin has aged slower than themselves. The paradox is false, simply because the twin on earth always remains in the same coordinate system, while the twin on the ship has to change to at least three coordinate systems (one travelling towards the destination at maximum speed, one doing a u-turn, and one travelling at maximum speed back to earth).

I also recommend a book on the theory of relativity written by Einstein himself. It was the most clear explanation of the theory I could find, and all of the ideas here are based on this book.

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